This study investigates sections of the vehicular impact scenario on reinforced concrete sections. A semi-trailer is considered as the vehicle for the impact event as this represents a worse-case scenario. Vehicle weight and impact velocity of the semi-trailer are considered as 80,000 lbs. (355.86 kN) and 100 ft./sec (30.48 m/sec) respectively to simulate a fully loaded trailer condition moving at its allowed speeds on the highway (Speeding and speed limits index and overview n.d.).

As a result of the parameters selected in this investigation, the vehicular impact scenario being studied is of the high-velocity, low duration variety. This will result in a high strain rate of loading for both the concrete and steel components of the RC pier. Concrete and steel both manifest a peculiar phenomenon when placed under such high loading rates (Auyeung et al. 2019). This phenomenon is an observable increase in strength capacity of these materials, as a function of the strain rate effect on reinforced concrete. As a result, a dynamic impact factor was proposed by (Malvar 1998), to reflect this increase on the strength capacity parameters of the reinforced concrete member.

### 4.1 Determination of dynamic increase factor (DIF)

Determination of the dynamic increase factor (DIF) reflecting the material behavior of concrete and steel during a vehicular impact scenario, involves an estimation of the dynamic flow stress from the impact. This dynamic flow stress can be estimated for either material. The dynamic flow stress (σ_{dyn}) in steel at impact is selected for use as it is projected to be the critical component of the member with respect to the coupler section. This dynamic flow stress is determined using Eq. 8 (Feyerabend 1988).

$$\sigma_{dyn}=\sigma_y\left[1+\left(\frac{\overset\,{\displaystyle\overset{\prime}\varepsilon}}C\right)^\frac1p\right]$$

(8)

Where: *σ*_{y} is a static flow stress and is considered as 60 ksi (420 MPa) for ASTM 706 Grade 60 steel rebar, *C* and *p* are the material constants (Cowper and Symonds 1957; Zhou and Li 2018) with values of 40 and 5 [8] respectively. Quasi-static strain rate of steel re-bar (έ) is considered as 0.16 sec^{− 1} for impacting velocity 100 ft./sec (30.48 m/sec) (Cowper and Symonds 1957).

The dynamic parameter ‘ξ’ can be computed from Eq. 9 (Malvar and Crawford 1998; Mander et al. 1988) and Eq. 8.

$$\xi =0.019-0.009\left(\frac{\sigma_{dyn}}{60}\right)$$

(9)

Where: *ξ* is a constant which depends on the dynamic yield stress of steel at the strain hardening zone, *έ* is the strain rate of steel and *σ*_{dyn} is the dynamic flow stress at uni-axial plastic strain rate of steel.

The DIF can be computed from Eq. 10 (Malvar and Crawford 1998; Mander et al. 1988), using Eq. 9.

$$DIF={\left(\frac{\overset{\prime{\mkern6mu}}{\varepsilon }}{10^{-4}}\right)}^{\xi }$$

(10)

Replacing the value of ξ from Eq. 9 to Eq. 10, yields a DIF of 1.053.

### 4.2 Computation of static force at impact

The time dependent frontal shock from vehicular impact can be computed using an averaged integration of the instantaneous impact force over the range of 50 ms near the peak impact force as shown in Eq. 11 (Zhou and Li 2018).

$${I}_{dyn}=\frac{\int_{t_d-0.025}^{t_d+0.025}{I}_r\sin \left(\frac{\pi {t}_d^{+}}{t}\right) di}{0.05}$$

(11)

Where: *I*_{dyn} represents the frontal shock due to impact, *I*_{r,} is the peak reflected pressure (overpressure), *t*_{d}^{+} is the time instant of the peak impact force, and *t* represents the impact duration.

Developed to estimate the total static force of a vehicle impact from the instantaneous peak force occurring during impact, the relationship shown in Eq. 11 captures the expected loading history of the impact over time on the RC pier (Zhou et al. 2017) utilizing the expected sinusoidal loading pattern of the impact event to predict the dynamic load from the peak force and the loading time history.

The overpressure represented by I_{r,} is a function of the kinetic energy (E) from the impacting vehicle and can be determined using Eq. 12, Eq. 12 was developed as a relationship between bending stress developed in the pier from the peak dynamic force of impact and kinetic energy using data from various simulated and experimental studies (Cao et al. 2019; Gomez and Alipour 2014; Mohammed and Parvin 2013; Zhou and Li 2018). The bending stress was used in lieu of the impact force so as to capture the possible effects of geometric variations of the pier in the resulting overpressure from regression analysis at impact as shown in Eq. 12 (Roy et al. 2021). Boundary conditions of the pier is considered as bottom end fixed and top end restrained from displacement and rotation (Zhang et al. 2018) is as shown in Fig. 4c.

$${I}_r=\left(4\times {10}^{-5}E\right)\ast \frac{4I}{(L.c)}$$

(12)

Where: *E* is the kinetic energy, absorbed by the impacted pier, *I* is the moment of inertia of the pier section, *L* is the height of the pier and *c* is the perpendicular distance from the neutral axis of the cross section to the farthest point on the cross section of the pier, as shown in Fig. 3.

Assuming the vehicle comes to rest without rebounding from the pier, the kinetic energy (E) equation is determined as same as the kinetic energy (E) of the vehicle using Eq. 13 (Tsang and Lam 2008).

$$E=0.5{M}_{veh}{V}^2$$

(13)

Where: *E* is impact energy of the vehicle, *M*_{veh} represents the weight of the impacting vehicle, and *V* is the frontal impact velocity of the vehicle causing instability of the column.

### 4.3 Computation of external flexural moments

Moments caused by the lateral impact force from vehicle collisions are induced at the base of the column as well as at different levels between the point of impact and the base. These moments if exceeding the moment capacity of the column could result in structural failure.

Assuming a pinned connection for the RC column, the static moment (M_{s}) induced by the vehicular collision can be determined as shown in Eq. 14.

$${M}_s={I}_{dyn}\ast H$$

(14)

Where: *H* is the height (in feet) to compute static moment and *I*_{dyn} represents the frontal shock due to impact.

#### 4.3.1 Dynamic moment

Dynamic moments (M_{dyn}) to investigate flexural effects at different fibers have been determined by static moment (M_{s}) times the dynamic increase factor (DIF). This is further expressed in Eq. 15 as to investigate the increased flexural effect (Feyerabend 1988).

$${M}_{dyn}= DIF\ast {M}_s$$

(15)

Where: *DIF* is the dynamic impact factor and *M*_{s} is the static moment.

### 4.4 Unbalanced moments from static and dynamic impact

Unbalanced moments in the column are caused by the exceedance of the moment capacity of the column and by the moments induced by the impact force. These unbalanced moments can be resisted by the splice-sleeve and grouted coupler system formed by the six couplers (six reinforcing steel bars are embedded into grout), one at the base of each longitudinal reinforcement as shown in the Fig. 4a. Determined as external static moments at column base for gross and core cross-sectional area minus sectional moments of resistance at gross and core cross-sectional areas, unbalanced static and dynamic moments for both gross and core cross-sectional areas are computed as shown in Eqs. 19 through 22. Figure 4a and b show the coupler arrangement in a column base embedded into the footing. Boundary conditions (fix-fix ended at bottom and top end is restrained from displacement and rotation) are shown in Fig. 4c. Computations of unbalanced moments are computed while semi-trailer impacts occur due collision at different heights in column has been further shown in Fig. 8.

#### 4.4.1 Unbalanced static moment for gross cross-sectional area

Static unbalanced moment (M_{un,sg}) for gross cross-sectional area has been computed in Eq. 16, as,

$${M}_{un, sg}={M}_{s,g}-{M}_{n,g}$$

(16)

Where: *M*_{s,g} is the external static moment for gross cross-sectional area.

#### 4.4.2 Unbalanced static moment for core concrete area

Static unbalanced moment (M_{un,sc}) for core cross-sectional area has been computed in Eq. 17, as,

$${M}_{un, sc}={M}_{s,c}-{M}_{n,c}$$

(17)

Where: *M*_{s,c} is the external static moment at core cross-sectional area of concrete column.

#### 4.4.3 Unbalanced dynamic moment for gross concrete area

Dynamic unbalanced moment (M_{un,dg}) for core cross-sectional area has been computed in Eq. 18, as,

$${M}_{un, dg}={M}_{dyn,g}-{M}_{n,g}$$

(18)

Where: M_{dyn,g} is the dynamic moment caused due to dynamic impact at gross cross-sectional area of concrete pier.

#### 4.4.4 Unbalanced dynamic moment for core concrete area

Dynamic unbalanced moment (M_{un,dc}) for core cross-sectional area has been computed in Eq. 19, and as shown in Fig. 5.

$${M}_{un, dc}={M}_{dyn,c}-{M}_{n,c}$$

(19)

Where: *M*_{dyn,c} is the dynamic moment caused due to impact at core cross-sectional area of concrete.